Two Types of Compound Inequalities

AND (intersection): $x$ must satisfy BOTH conditions.
Written as: $a < x < b$ or " $x > a$ AND $x < b$ "
Graph: a segment between two values.

1 ≤ x < 7 (AND compound inequality)

OR (union): $x$ must satisfy AT LEAST ONE condition.
Written as: " $x < a$ OR $x > b$ "
Graph: two rays going in opposite directions.

The SAT mostly tests AND (double inequalities).

Theory

Solving Double Inequalities

A double inequality like $3 < 2x + 1 < 11$ means:
$3 < 2x + 1 \quad \text{AND} \quad 2x + 1 < 11$

Shortcut: solve all three parts at once — apply the same operation to all three sections.

Example 1

Solve $3 < 2x + 1 < 11$

Subtract $1$ from all three parts:

$2 < 2x < 10$

Divide all three by $2$ :

$1 < x < 5$

1 < x < 5

Example 2

Solve $-6 \leq -3x < 12$

Divide all three by $-3$ (flip BOTH signs):

$2 \geq x > -4$

Rewrite in standard order:

$-4 < x \leq 2$

-4 < x \leq 2

2 more sections including examples, practice problems, and step-by-step solutions.

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